Extending Datatype Support in Web Ontology Reasoning

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1 Extending Datatype Support in Web Ontology Reasoning Jeff Z. Pan and Ian Horrocks Information Management Group Department of Computer Science University of Manchester Oxford Road, Manchester M13 9PL, UK Abstract. The Semantic Web is a vision of the next generation Web, in which semantic markup will make Web resources more accessible to automatic processes. Description Logics (DLs) are of crucial importance to the development of the Semantic Web, where their role is to provide formal underpinnings and automated reasoning services for Semantic Web ontology languages such as DAML+OIL. In this paper, we show how the description logic SHOQ(D), which has been designed to provide such services, can be extended with n-ary datatype predicates and qualified number restrictions with n-ary datatype predicates, to give SHOQ(D n), and we present an algorithm for deciding the satisfiability of SHOQ(D n) concepts, along with a proof of its soundness and completeness. The work is motivated by the requirement for n-ary datatype predicates and qualified number restrictions with n-ary predicates in relation to real world properties in semantic Web ontologies and applications. 1 Introduction The Semantic Web [2] is a vision of the next generation Web,in which the current rendering markup,which specifies how to display Web resources for human consumption,will be enhanced with so called semantic markup,which will specify the meaning of web resources so as to make them more accessible to automatic processes. Description Logics (DLs) are of crucial importance to the development of the Semantic Web,where their role is to provide formal underpinnings and automated reasoning services for semantic Web ontology languages [3,11] such as DAML+OIL 1 [7,12]. Significant effort has already been devoted to the investigation of suitable DLs in particular,[8] have presented the SHOQ(D) DL, along with a sound and complete algorithm for deciding concept satisfiability,a basic reasoning service for DLs and ontologies. A key feature of SHOQ(D) is that,like DAML+OIL,it supports datatypes [1,8,9] (e.g.,string,integer) as well as the usual abstract concepts 1 R. Meersman, Z. Tari (Eds.): CoopIS/DOA/ODBASE 2002, LNCS 2519, pp , c Springer-Verlag Berlin Heidelberg 2002

2 1068 J.Z. Pan and I. Horrocks (e.g.,animal,plant). SHOQ(D),however,supports a very restricted form of datatypes,i.e.,it can only deal with unary datatype predicates. While this is quite close 2 to the requirements of the current version of the DAML+OIL language,it is not enough for (even the current version of) DAML+OIL and some semantic Web ontologies and applications. An approach of extending DL with datatypes was first introduced by [1],who described a datatype (D) extension of the well known ALC DL. [1] have shown that although the satisfiability of ALC(D) is decidable,if ALC(D) is extended with transitive closure of features,the satisfiability problem is undecidable. Lutz [9] proved that reasoning with ALC(D) and general TBoxes is undecidable. In order to extend expressive DLs with concrete domains,horrocks and Satler [8] proposed a simplified approach on concrete domain and applied this approach on the SHOQ(D) DL. Pan [10] investigated the simplifying constraints of SHOQ(D) w.r.t. datatypes,and showed how these could be relaxed in order to extend SHOQ(D) with n-ary datatype predicates. We should mention that, similar to Baader and Hanschke [1] s approach,haarslev et al. [5] extended the SHN DL with restricted concrete domain (D) and gave the SHN(D) DL, which supports n-ary datatype predicates without qualified number restrictions. In this paper,we extend our work in [10] and add qualified number restrictions with n-ary predicates to give the SHOQ(D n ) DL,and we present a sound and complete decision procedure for deciding concept satisfiability and subsumption in this logic. The rest of the paper is organised as follows. In Section 2,we introduce some basic concepts of datatypes and give some concrete examples to demonstrate why n-ary datatype predicates,as well as qualified number restrictions with n-ary datatype predicates,are often necessary in semantic Web ontologies and applications. Section 3 briefly compares the two approaches to extend description logics with datatypes. In Section 4,we introduce SHOQ(D n ), which supports the n-ary datatype predicates and qualified number restrictions with n-ary datatype predicates. In Section 5,we give a tableau algorithm for SHOQ(D n ) and its decidability proof. Section 6 is a brief discussion of possible future work. 2 Datatypes in the Semantic Web Datatypes are important in semantic Web ontologies and applications,because most of which need to represent,in some way,various real world properties such as size,weight and duration,and some other complex user defined datatypes. Reasoning and querying over datatype properties are important and necessary if these properties are to be understood by machines. Before we go further,we introduce the following basic concepts (about datatype),which we will use in this paper: 2 DAML+OIL supports unary datatype predicates and qualified number restrictions with unary datatype predicates.

3 Extending Datatype Support in Web Ontology Reasoning 1069 Datatype is a declaration of the kind of data,i.e. integer and string. The datatype determines which constraints/operations can be performed on such data values. Datatype properties can be used to represent real world properties,i.e. size,weight and duration. They have data values,which belong to certain datatypes. Datatype predicates represent constraints over a list of datatype properties. Usually we call them n-ary datatype predicates if the constraints are defined over n datatype properties. Currently DAML+OIL supports only unary datatype predicates and qualified number restrictions on unary datatype predicates,e.g. you can define a less than 21 years old predicate over datatype property age. Many people think it is not enough,while n-ary datatype predicates and qualified number restrictions on n-ary predicates are often necessary. Here are some concrete examples: Example 1 E-ontologies may need to classify items according to their sizes, and to reason that an item which has height less than 5cm and the sum of length and width less than 10cm is a kind of items for which no shipping costs are charged. Here height,length and width are datatype properties and the sum of... less than 10cm is an n-ary datatype predicate. Example 2 Let s go back to the age example. Suppose in an ontology we define a class Person with a datatype property age. If one of its instance is John, and John has age 20,then John s age satisfies the predicate less than 21 years old. This may be true now,but what about in 10 years is it still true? Of course not,because our ages change every year. One solution is to define three datatype properties birth-year, current-year and real age and to define a minus predicate over them,which is 3-ary,so that we can calculate the real age by birth-year and current-year. Example 3 E-ontologies may need to classify customers according to the numbers of their friends addresses they provide,and decide that those who provide (1) at least 10 friends e-addresses,(2) among them at least 5 friends e-addresses are from UK (e.g. man.ac.uk),(3) and at least 5 friends e-addresses have the same domain (e.g. hotmail.com) as their e-addresses can have 5% cash back during promotion. Here friends e-addresses and their e-addresses are datatype properties,and have the same domain as is an n- ary datatype predicate. Qualified number restrictions with n-ary predicates are also used in this example. They are very expressive in the sense that they force restrictions on the numbers of the tuples of data values that satisfy the datatype predicate,instead of on the numbers of a certain datatype property. Therefore they are quite different from the qualified number restrictions that we have in the SHOQ(D) DL and become new challenges to DL reasoners. (See section 5 to find out how we solve this problem.) From these examples,we can see n-ary datatype predicates and qualified number restrictions with n-ary predicates are very important and useful in semantic Web ontologies and applications. On the one hand,it is very desirable

4 1070 J.Z. Pan and I. Horrocks to express business rules [4] or constraints for datatype properties in semantic Web ontologies and applications. On the other hand,too expressive rules and constraints could easily introduce undecidability. In this paper,we prove that adding n-ary datatype predicates as well as qualified number restrictions with n-ary predicates to the SHOQ(D) DL,which is quite suited to the provision of reasoning support for DAML+OIL,does not change the decidability property of the SHOQ(D) DL. Thus we suggest that future versions of DAML+OIL, or OWL 3 can include these extended datatype constructs without worrying the decidability problem. 3 Description Logics and Datatypes For many knowledge representation application,it is essential to integrate reasoning about datatype properties with reasoning about knowledge represented at the abstract level. One theoretically well-founded approach,first described by Baader and Hanschke [1] is to extend a DL with concrete domains. Concrete domains were defined as follows: Definition 1 A concrete domain is a pair ( D,Φ D ), where D is a set called the domain and Φ D is a set of predicate names. Each predicate name P is associated with an arity n, and an n-ary predicate P D dom(d) n. A concrete domain is called admissible iff 1. the set of its predicate names is closed under negation and contains a name D for D, and 2. the satisfiability problem for an expression of the form P D 1... P D n is decidable. In order to extend expressive DLs with concrete domains,horrocks and Sattler [8] proposed a simplified approach on concrete domain and give the SHOQ(D) DL. Pan [10] investigated the simplifying constraints introduced in [8] and extend the SHOQ(D) DL with n-ary predicates. One of the simplifying constraints about concrete datatype is that D is defined as the domain of all concrete datatypes,instead of any specific concrete domain D. Here we give a formal definition of (some of) the concepts we introduced in Section 2. Definition 2 D is the datatype domain covering all concrete datatypes. A set of datatypes d 1...,d m can be defined in D, whose domain are subsets of the datatype domain, i :1 i m, d D i D. Datatype predicates P 1,...,P r can be defined over these datatypes, and each predicate P j is associated with an arity n j, where j :1 j r, Pj D d D j1... dd jn j nj D. Each such datatype predicate P j (v 1,...,v nj ) is a boolean function, where v 1,...,v nj are its input variables. It returns true iff there exists a tuple of data values t 1,...,t nj in D, s.t. i :1 i n j,vi D = t i and t 1,...,t nj Pj D. 3

5 Extending Datatype Support in Web Ontology Reasoning 1071 The negation of datatype predicate P j (v 1,...,v nj ), i.e. P j (v 1,...,v nj ),is also a boolean function. It returns true iff. P j (v 1,...,v nj ) returns false. An example of datatype is INT+,which has the set of nonnegative integers as its domain. A binary predicate and a unary predicate n are defined over INT+. Definition 3 A set of datatypes d 1...,d m is conforming if 1. the datatype domain D is disjoint with the object domain I, and 2. a required binary inequality-checking predicate i {P 1,...,P r }, is defined for each datatype d i, and 3. there exists a sound and complete decision procedure for checking if a set of tuples of data values satisfy a set of related datatype predicates P 1,...,P r. The datatype INT+ is not conforming,since it doesn t have a built-in binary inequality-checking predicate. Thus any set of datatypes which has INT+ as its member is not conforming due to the second condition in Definition 3. Please note that according to Definition 2,datatype predicates can be defined over different datatypes,and Definition 3 requires that each conforming datatype have a built-in binary inequality-checking predicate. In the simplified approach, type systems are introduced to define datatypes and datatype predicates,including the built-in binary inequality-checking predicates for all supported data- types. Type systems are used with DL reasoners, and can answer boolean questions,e.g. to check if a set of tuples of data values satisfy a set of predicates simultaneously. With type systems,we can deal with an arbitrary conforming set of datatypes and predicates without compromising the compactness of the concept language or the soundness and completeness of our decision procedure [8]. Another main difference between the two approaches,besides the definition of the datatype domain,is that in the new approach,datatype properties are roles instead of features (functional roles),which makes it possible to have qualified number restrictions on n-ary predicates while with features,the number restriction is already set as less than or equal to 1. In next section,we will give the definition of the SHOQ(D n ) DL. Please note that in DLs, classes are usually called concepts, properties are often called roles,and datatype properties are also called concrete roles. 4 SHOQ(D n ) Definition 4 Let C, R = R A R D, I be disjoint sets of concept, abstract and concrete role and individual names. For R and S roles, a role axiom is either a role inclusion, which is of the form R S for R, S R A or R, S R D,or a transitivity axiom, which is of the form Trans(R) for R R A.Arole box R is a finite set of role axioms. A role R is called simple if, for * the transitive reflexive closure of on R and for each role S, S * R implies Trans(S) R.

6 1072 J.Z. Pan and I. Horrocks The set of concept terms of SHOQ(D n ) is inductively defined. As a starting point of the induction, any element of C is a concept term (atomic terms). Now let C and D be concept terms, o be an individual, R be a abstract role name, T 1,...,T n be concrete role names, P be an n-ary datatype predicate name. Then the following expressions are also concept terms: 1. (universal concept) and D (universal datatype), 2. C D (disjunction), C D (conjunction), C (negation), and {o} (nominals), 3. R.C (exists-in restriction) and R.C (value restriction), 4. mr.c (atleast restriction) and mr.c (atmost restriction), 5. T 1,,T n.p n (datatype exists) and T 1,,T n.p n (datatype value), 6. mt 1,...,T n.p n (datatype atleast) and mt 1,...,T n.p n (datatype atmost), 7. mt (concrete role atleast), mt (concrete role atmost). SHOQ(D n ) extends SHOQ(D) by supporting n-ary datatype predicates P n. The interpretations of datatype constructs are listed in Figure 1. Note that concrete role atleast (atmost) is only a special form of datatype atleast (atmost, respectively) where n = 1 and P n = D. To illustrate the use of SHOQ(D n )-concept,let s go back to the examples we used in Section 2: 1. Items with height less than 5cm,and the sum of their length and width less that 10cm can be defined as a SHOQ(D n )-concept Item =1height.< 5cm =1length =1width length, width.sum < 10cm where =1 is a shortcut for 1 1,and height, length and width are concrete roles, < 5cm is a unary datatype predicate and sum < 10cm is a binary predicate. Note that < 5cm and sum < 10cm are datatype predicates,rather than datatype atmost. 2. Persons who are younger than 21 years old can be defined as a SHOQ(D n )- concept P erson =1birth year =1current year =1age.< 21 birth year, current date, age.minus where minus is a trinary datatype predicate,which can be defined in the type system as lambda expression λ(x, y, z)(z = y x). 3. Customers in example 3 can be defined as a SHOQ(D n )-concept Customer 10friends eaddress 5friends eaddress.from UK 5friends eaddress, eaddress.same domain where 10friends eaddress is a concrete role atleast,and 5friends eaddress.from UK and 5friends eaddress, eaddress.same domain are datatype atleast. Now we define a tableau for SHOQ(D n ). For ease of presentation,we assume all concepts to be in negation normal form (NNF). We use C to denote the NNF of C. Moreover,for a concept D,we use cl(d) to denote the set of all sub-concepts of D,the NNF of these sub-concepts,and the (possibly negated) datatypes occurring in these (NNF of) sub-concepts.

7 Extending Datatype Support in Web Ontology Reasoning 1073 Construct Name Syntax Semantics universal datatype D D D = D datatype exists T 1,,T n.p n ( T 1,,T n.p n) I = {x I y 1 y n. x, y 1 T1 I x, y n Tn I y 1, y n Pn D } datatype value T 1,,T n.p n ( T 1,,T n.p n) I = {x I y 1 y n. x, y 1 T1 I x, y n Tn I y 1, y n Pn D } datatype atleast mt 1,...,T n.p n ( mt 1,...,T n.p n) I = {x I { y 1 y n x, y 1 T1 I x, y n Tn I y 1, y n Pn D } m} datatype atmost mt 1,...,T n.p n ( mt 1,...,T n.p n) I = {x I { y 1 y n x, y 1 T1 I x, y n Tn I y 1, y n Pn D } m} concrete role atleast mt ( mt ) I = {x I {y D x, y T } m} concrete role atmost mt ( mt ) I = {x I {y D x, y T } m} Fig. 1. Datatype constructs in SHOQ(D n) Definition 5 If D is a SHOQ(D n )-concept in NNF, R a role box, and R D A, R D D are the sets of abstract and concrete roles occurring in D or R, a tableau T for D w.r.t. R is defined as a quadruple (S, L, E A, E D ) such that: S is a set of individuals, L : S 2 cl(d) maps each individual to a set of concepts which is a subset of cl(d), E A : R D A 2S S maps each abstract role in R D A to a set of pairs of individuals, E D : R D D 2S D maps each concrete role in R D D to a set of pairs of individuals and concrete values, and there is some individual s S such that D L(s). For all s, t S, C, C 1,C 2 cl(d), R, S R D A, T,T,T 1,...,T n R D D, n-ary predicate P n and S T (s, C) :={t S s, t E A(S) and C L(t)}, T 1T 2...Tn T (s, P n):={ y 1,...,y n Pn D s, y 1 E D(T 1),..., s, y n E D(T n)}, true if s, y i E D(T n)(1 i n) and DC T (s, T 1,...,T n,y 1,...,y n,p n):= y 1,...,y n Pn D false otherwise it holds that: (P1) if C L(s), then C / L(s), (P2 ) if C 1 C 2 L(s), then C 1 L(s) and C 2 L(s), (P3) if C 1 C 2 L(s), then C 1 L(s) or C 2 L(s), (P4) if s, t E A (R) and R * S, then s, t E A (S), if s, t E D (T ) and T * T, then s, t E D (T ), (P5) if R.C L(s) and s, t E A (R), then C L(t), (P6) if R.C L(s), then there is some t S such that s, t E A (R) and C L(t), (P7) if S.C L(s) and s, t E A (R) for some R * S with Trans(R), then R.C L(t), (P8) if ns.c L(s), then S T (s, C) n, (P9) if ns.c L(s), then S T (s, C) n, (P10) if { ns.c, ns.c} L(s) and s, t E A (S), then {C, C} L(t), (P11) if {o} L(s) L(t), then s = t,

8 1074 J.Z. Pan and I. Horrocks (P12) (P13) if T 1,,T n.p n L(s) and s, t 1 E D (T 1 ),, s, t n E D (T n ), then DC T (s, T 1,...,T n, t 1,...,t n,p n )=true, if T 1,,T n.p n L(s), then there is some t 1,,t n D such that s, t 1 E D (T 1 ),, s, t n E D (T n ), DC T (s, T 1,...,T n,t 1,...,t n,p n )=true, (P14) if mt 1,...,T n.p n L(s), then T 1 T 2...Tn T (s, P n ) m, (P15) if mt 1,...,T n.p n L(s), then T 1 T 2...Tn T (s, P n ) m, (P16) if { mt 1,...,T n.p n, mt 1,...,T n.p n } L(s) and s, t 1 E D (T 1 ),..., s, t n E D (T n ), then for 1 i n, we have either DC T (s, T 1,...,T n,t 1,...,t n,p n ) = true, or DC T (s, T 1,...,T n,t 1,...,t n, P n )=true. Lemma 1. A SHOQ(D n )-concept D in NNF is satisfiable w.r.t. a role box R iff D has a tableau w.r.t. R. Proof: For the if direction,if T =(S, L, E A, E D ) is a tableau for D, amodel I =( I, I) ofd can be defined as: I = S, CN I = {s CN L(s)} for all concept names CN in cl(d), if R R +, R I A = E A(R) +, otherwise R I A = E A(R) P I, P * R,P R R I D = E D(R),where E A (R) + denotes the transitive closure of E A (R). D I because s 0 D I. Here we only concentrate on (P14) to (P15); the remainder is similar to the proofs found in [10] E = mt 1,...,T n.p n. According to (P14), E L(s) implies that T 1T 2...T T n (s, P n) m. By the definition of T 1T 2...T T n (s, P n), we have s {x I { t 1,...,t n x, t 1 E D (T 1)... x, t n E D (T n) t 1,...,t n P D n } m}, Since E D (T i) = T I i, we have s ( mt 1,...,T n.p n) I. Similarly, if E = mt 1,...,T n.p n,wehaves ( mt 1,...,T n.p n) I. For the converse,if I = ( I, I) isamodelofd,then a tableau T = (S, L, E A, E D ) for D can be defined as: S = I, E A (R) = RA I, E D(R) = RD I, L(s) ={C cl(d) s CI }. It only remains to demonstrate that T is a tableau for D: T satisfies (P14) to (P16) as a direct consequence of the semantics of datatype constructs. 5 A Tableau Algorithm for SHOQ(D n ) Form Lemma 1,an algorithm which constructs a tableau for a SHOQ(D n )- concept D can be used as a decision procedure for the satisfiability of D with respect to a role box R. 4 Note that in this paper, we mainly focus on the proof of the number restriction on concrete roles, the remainder is similar to the proofs found in [10].

9 Extending Datatype Support in Web Ontology Reasoning 1075 Definition 6 Let R be a role box, D a SHOQ(D n )-concept in NNF, R D A the set of abstract roles occurring in D or R, and I D the set of nominal occurring in D. A tableaux algorithm works on a completion forest for D w.r.t. R, which is a set of trees F. Each node x of the forest is labelled with a set L(x) cl(d) { (R, {o}) R R D A and {o} I D }, and each edge x, y is labelled with a set of role names L( x, y ) containing roles occurring in cl(d) orr. Anodex is either an abstract node or a concrete node, the latter one represents a data value, and is always a leaf of F. Additionally, we keep track of inequalities between nodes of the tree with a symmetric binary relation, which can be used between two abstract nodes, or between two tuples of concrete nodes of F. Inequality relations between tuples of concrete nodes can be mapped to inequality-checking predicates provided by the type systems 5.For each {o} I D there is a distinguished node x {o} in F such that {o} L(x). The algorithm expands the forest either by extending L(x) for some node x or by adding new leaf nodes. Given a completion forest, a node y is called an R-successor ofanodex if, for some R with R * R, either y is a successor of x and R L( x, y ), or (R, {o}) L(x) and y = x {o}. Ancestors and roots are defined as usual. For an abstract role S andanodex in F we define S F (x, C) by S F (x, C) :={y y is an S-successor of x and C L(y)}. Given a completion forest, concrete nodes t 1,...,t n are called T 1 T 2...T n - successors of a node x if, for some concrete rolest 1,...,T n with T i * T i, t 1,...,t n are successors of x and T i L( x, t i ), 1 i n. Foranodex, its T 1 T 2...T n -successors t 1,...,t n, n-ary datatype predicate P n, we define a set DC F by DC F = {< DCelement >} where DC F is a set of DCelements, which have the form < DCelement >= {x, T 1,...,T n, t 1,...,t n,p n} DC F is initialised as an empty set. DC F is is satisfied iff. (i) there exists value : N C D, where N C is the set of all concrete nodes, s.t. for all {x, T 1,...,T n, t 1,...,t n,p n } DC F, value(t 1 ),...,value(t n ) P D n are true; (ii) the inequality relations between tuples of concrete nodes are satisfied. In order to retrieve the set of all the T 1 T 2...T n -successors of x, which satisfy a certain predicate P n, we define DCSuccessors F (x, P n ) by DCSuccessors F (x, T 1,...,T n,p n):={ t 1,...,t n {x, T 1,...,T n, t 1,...,t n,p n} DC F } In order to retrieve the set of datatype predicates, which are satisfied by T 1 T 2... T n successors t 1,...,t n of x, we define DCP redicates F (x, T 1,...,T n,t 1,...,t n ) by 5 See the definition of conforming datatypes in section 3

10 1076 J.Z. Pan and I. Horrocks DCP redicates F (x, T 1,...,T n,t 1,...,t n):={p n {x, T 1,...,T n, t 1,...,t n,p n} DC F } Note that we can use * as parameter in DCSuccessors F and DCP redicates F, e.g. DCSuccess ors F (x,, ) means all the concrete successors of node x. Anodex is directly blocked if none of its ancestors are blocked, and it has an ancestor x that is not distinguished such that L(x) L(x ).Wecallx blocks x. Anodeisblocked if it is directly blocks or if its predecessor is blocked. If {o 1 },,{o l } are all individuals occurring in D, the algorithm initialises the completion forest F to contain l +1 root nodes x 0,x {o1},,x {ol } with L(x 0 )={D} and L(x {oi}) ={{o i }}. The inequality relation is initialised with the empty relation. F is then expended by repeatedly applying the expansion rules, listed in Figure 5 6, stopping if a clash occurs in one of its nodes. Foranodex, L(x) is said to contain a clash if: 1. for some concept name A N C, {A, A} L(x), or 2. for some role S, S.C L(x) and there are n +1 S-successors y 0,,y n of x with C L(y i ) for each 0 i n and y i y j for each 0 i<j n, or 3. DC F isn t satisfied; 6 Figure 5 only lists the rules about datatypes, other rules can be found in [10].

11 Extending Datatype Support in Web Ontology Reasoning for some concrete roles T 1,...,T n, n-ary datatype predicate P n, mt 1,...,T n. P n L(x), we have DCSuccessors F (x, T 1,...,T n,p n ) m + 1, or 5. for some {o} L(x),x x {o}. The completion forest is complete when, for some node x, L(x) contains a clash, or when none of the expansion rules is applicable. If the expansion rules can be applied in such a way that they yield a complete, clash-free completion forest, then the algorithm returns D is satisfiable w.r.t. R, and D is unsatisfiable w.r.t. R otherwise. Lemma 2. (Termination) When started with a SHOQ(D n )-concept D in NNF, the tableau algorithm terminates. Proof: Let d = cl(d), k = R D A, n max the maximal number in atleast number restrictions as well as datatype atleast,and l = I D. Here we mainly concentrate on rules about number restriction on concrete roles. Note that the set of datatypes used is conforming,so we don t have to worry about the checking on concrete nodes. Termination is a consequence of the following properties of the expansion rules: 1. Each rule but the -, P -ortheo-rule strictly extends the completion forest,by extending node labels or adding nodes,while removing neither nodes nor elements from node. 2. New nodes are only generated by the -, P -, -rule or the P -rule as successors of a node x for concepts of the form R.C, T 1,,T n.p n, ns.c and mt 1,,T n.p n in L(x).Foranodex,each of these concepts can trigger the generation of successors at most once even though the node(s) generated was later removed by either the -, P -ortheo-rule. For the P -rule: If T 1 T 2...T n -successors t 11,,t 1n,, t m1,,t mn were generated by an application of the P -rule for a concept ( mt 1,,T n.p n ),. then t j1,,t jn = t k1,,t kn holds for all 1 i n and 1 j < k m. This implies there will always be m T 1 T 2...T n -successors. t 11,,t 1n,, t m1,,t mn of x with P n (i) L(t i ) and t j1,,t jn = t k1,,t kn holds for all 1 i n and 1 j < k m,since the -, O- and P -rule can never merge them,and,whenever an application of the P -rule sets some L(t ji )to,then there will be some T 1 T 2...T n -successors t k1,,t kn of x with P n (i) L(t ki ) and t k1,,t kn inherits all inequalities from t j1,,t jn. Hence the out-degree of the forest is bounded by d n max. 3. Nodes are labelled with subsets of cl(d) { (R, {o}) R R D A and {o} I D },and the concrete value nodes are always leaves,so there are at most 2 d+kl different node labellings. Therefore,if a path p is of length at least 2 d+kl,then,from the blocking condition above,there are two nodes x, y on p such that x is directly blocked by y. Hence paths are of length at most 2 d+kl.

12 1078 J.Z. Pan and I. Horrocks Lemma 3. (Soundness) If the expansion rules can be applied to a SHOQ(D n )-concept D in NNF and a role box R such that they yield a complete and clash-free completion forest, then D has a tableau w.r.t. R. Proof: Let F be the complete and clash-free completion forest constructed by the tableaux algorithm for D. To cope with cycle,an individual in S corresponds to a path in F. Due to qualifying number restrictions,we must distinguish different nodes that are blocked by the same node. We refer the readers to [10] for the definitions of path and related concepts. We can define a tableau T =(S, L, E A, E D ) with: S = Paths(F), L(p) = L(Tail(p)), E A (R A ) = { p, q S S q =[p (x, x )] and x is an R A -successor of Tail(p), } E D (R D )={ p, value(t) S D t is an R D -successor of Tail(p)}. We have to show that T satisfies (P14) to (P16) from Definition 5. (P14): Assume mt 1,...,T n.p n L(p). This implies that in F there exist mt 1 T 2...T n -successors t 11,...,t 1n,..., t m1,...,t mn of Tail(p),s.t.. t j1,,t jn = t k1,,t kn and {p, T 1,...,T n, t j1,...,t jn,p n } DC F for all 1 j<k m (otherwise, P -rule was still applicable). We claim that,for each of these concrete nodes,according to the construction of E D above,we have p, value(t ji ) E D (T i ) and value(t j1 ),...,value(t jn ) Pn D for all 1 i n and 1 j m. According to the satisfiability of DC F and definition of T 1 T 2...Tn T (p, P n ),this implies T 1 T 2...Tn T (p, P n ) m. (P15): Assume (P15) doesn t hold. Hence there is some p S with ( mt 1,...,T n.p n ) L(p) and T 1 T 2...Tn T (p, P n ) > m. According to the definition of T 1 T 2...Tn T (p, P n ),this implies that there exist t 11,...,t 1n,..., t m+1,1,...,t m+1,n such that p, value(t ji ) E D (T i ), and {p, T 1,...,T n, t j1,...,t jn,p n } DC F,for all 1 i n and 1 j < k m + 1. Since the m + 1 tuples of data values aren t equal to each other,so the m + 1 tuples of concrete nodes aren t equal to each other. Thus DCSuccessors F (x, T 1,...,T n,p n ) m + 1,which fires a clash. Thus the assumption T 1 T 2...Tn T (p, P n ) >mis false. So we have T 1 T 2...Tn T (p, P n ) m. (P16): Assume { mt 1,,T n.p n, mt 1,,T n.p n } L(p), p, t i E D (T i ), 1 i n,thus t 1,...,t n is a T 1 T 2...T n -successors of Tail(p). Let value(t i ) be the value of t i : (1) if {p, T 1,...,T n, t 1,...,t n,p n } DC F, we have DC T (p, T 1,...,T n, value(t 1 ),...,value(t n ), P n ) = true; (2) if {p, T 1,...,T n, t 1,...,t n, P n } DC F,we have DC T (p, T 1,...,T n, value(t 1 ),...,value(t n ), P n )=true. Lemma 4. (Completeness) If a SHOQ(D n )-concept D in NNF has a tableau w.r.t. R, then the expansion rules can be applied to D and R such that they yield a complete, clash-free completion forest. Proof: Let T = (S, L, E A, E D ) be a tableau for D w.r.t. a role box R. We use T to guide the application of the non-deterministic rules. We define a function π,mapping the nodes of the forest F to S D such that

13 Extending Datatype Support in Web Ontology Reasoning 1079 L(x) L(π(x)); π(x),π(y) E A if: 1. π(y) S and y is an R A -successor of x,or 2. (R, {o}) L(x) and y = x {o} ; π(x),π(y) E D if π(y). D and y is an R D -successor of x; abstract nodes x = y implies π(x) π(y); while tuples of concrete nodes y j1,...,y jn yk1,...,y kn implies π(y j1 ),...,π(y jn ) π(y k1 ),...,π(y kn ). ( ) We only have to consider the various rules about number restriction on datatypes. The P -rule: If mt 1,...,T n.p n L(x),then mt 1,...,T n.p n L(π(x)). Since T is a tableau,(p14) of Definition 5 implies that T 1 T 2...Tn T (π(x),p n ) m. Hence there are m tuples of data values. t 11,...,t 1n,..., t m1,...,t mn,such that π(x),t ji E D, t j1,...,t jn = t k1,...,t kn,and DC T (π(x),t 1,...,T n,t j1,...,t jn,p n )=true,for 1 i n and 1 j<k m. The P -rule generates m new T 1 T 2...T n -successors y 11,...,y 1n,..., y m1,...,y mn. By setting π := π[y ji t ji ](1 i n, 1 j < k m),one obtains a function π that satisfies ( ) for the modified forest. The P -rule: If mt 1,...,T n.p n L(x),then mt 1,...,T n.p n L(π(x)). Since T is a tableau,(p15) of Definition 5 implies T 1 T 2...Tn T (π(x),p n ) m. If the P -rule is applicable,we have DCSuccessors F (x, T 1,...,T n,p n ) > m,which implies that there are at least m +1 T 1 T 2...T n -successors y 11,...,y 1n,..., y m+1,1,...,y m+1,n such that {x, T 1,...,T n, y j1,...,y jn,p n } DC F,for 1 j m + 1. Thus,there must be two y j1,...,y jn and y k1,...,y kn among the m+1 T 1 T 2...T n -successors such that π(y j1 ),...,π(y jn ) = π(y k1,...,π(y kn (otherwise T 1 T 2...Tn T (π(x),p n ) >mwould hold). This implies y j1,...,y jn y k1,...,y kn cannot hold because of ( ). Hence the P -rule can be applied without violating ( ). The choose P -rule: If { mt 1,,T n.p n, mt 1,,T n.p n } L(x),we have { mt 1,, T n.p n, mt 1,,T n.p n } L(π(x)),and if there are T 1 T 2...T n - successors y 1,...,y n of x,then π(x),π(y i ) E D, 1 i n,due to ( ). Since T is a tableau,(p16) of Definition 5 implies either DC T (π(x),t 1,...,T n,π(y 1 ),...,π(y n ),P n ) = true,or DC T (π(x),t 1,...,T n,π(y 1 ),..., π(y n ), P n ) = true. Hence the choose P -rule can accordingly either set DC F DC F {x, T 1,... T n, y 1,...,y n,p n },or set DC F DC F {x, T 1,...,T n, y 1,...,y n, P n }. Whenever a rule is applicable to F,it can be applied in a way that maintains ( ),and,from Lemma 2,we have that any sequence of rule applications must terminate. Since ( ) holds,any forest generated by these rule-applications must be clash-free. This can be seen from the condition described in [8] plus the following:

14 1080 J.Z. Pan and I. Horrocks If F does not satisfy DC F,there must be some concrete nodes from which no values mapping satisfies all the relevant predicates,and therefore there can be no values satisfying all of properties (P12) to (P16). F cannot contain a node x with mt 1,...,T n.p n L(x),and m + 1 T 1 T 2...T n -successors t 11,...,t 1n,..., t m+1,1,...,t m+1,n of x with t j1,...,t jn t k1,...,t kn and {x, T 1,...,T n, t j1,...,t jn,p n } DC F, for all 1 i n, 1 j < k m + 1,because mt 1,...,T n.p n L(π(x)),and,since t j1,...,t jn t k1,...,t kn implies π(t j1 ),...,π(t jn ) π(t k1 ),...,π(t kn ), T 1 T 2...Tn T (π(x),p n ) >mwould hold which contradicts (P15) of Definition 5. As an immediate consequence of Lemmas 2,4,5 and 6, the completion algorithm always terminates,and answers with D is satisfiable w.r.t. R iff. D has a tableau T. Next,subsumption can be reduced to (un)satisfiability. Finally, SHOQ(D n ) can internalise general concept inclusion axions [6]. However,in the presence of nominals,we must also add O.o 1 O.o l to the concept internalising the general concept inclusion axioms to make sure that the universal role O indeed reaches all nominals O i occurring in the input concept and terminology. Thus,we can decide these inference problems also w.r.t. terminologies. Theorem 7 The tableau algorithm presented in Definition 6 is a decision procedure for satisfiability and subsumption of SHOQ(D n )-concepts w.r.t. terminologies. 6 Discussion As we have seen,unary datatype predicates are usually not enough,while n-ary datatype predicates as well as qualified number restrictions with n-ary datatype predicates are often necessary when modelling the concrete properties of real world entities in semantic Web ontologies and applications. Therefore,we have extended SHOQ(D) with n-ary datatype predicates and qualified number restrictions with n-ary predicates to give the SHOQ(D n ) DL. We have shown that the decision procedure for concept satisfiability and subsumption is still decidable in SHOQ(D n ). An implementation based on the FaCT system is planned, and will be used to test empirical performance. With its support for nominals,n-ary datatype predicates and qualified number restrictions with n-ary datatype predicates, SHOQ(D n ) is well suited to provide reasoning support for ontology languages in general,and semantic Web ontology languages in particular. Therefore,we think that future versions of Web ontology languages,e.g. DAML+OIL and OWL,can include n-ary datatype predicates and qualified number restrictions with n-ary datatype predicates. As future work,it is interesting to study how the extended datatype support introduced in the paper can help to express various business rules required in semantic Web ontologies and applications. Secondly,it is also important to extend current optimisation techniques to cope with nominals used in the SHOQ(D n ) DL.

15 Extending Datatype Support in Web Ontology Reasoning 1081 Acknowledgements. We would like to thank Ulrike Sattler,since the work presented here extends the original work on SHOQ(D). Thanks are also due to Carsten Lutz and Volker Haarslev for their helpful discussion on concrete datatypes. References 1. F. Baader and Philipp Hanschke. A Scheme for Integrating Concrete Domains into Concept Languages. In IJCAI-91, pages , Tim Berners-lee. Semantic Web Road Map. W3C Design Issues. Oct J. Broekstra, M. Klein, S. Decker, D. Fensel, F. can Harmelen, and I. Horrocks. Enabling knowledge representation on the Web by extending RDF Schema, Nov Benjamin N. Grosof and Terrence C. Poon. Representing Agent Contracts with Exceptions using XML Rules, Ontologies, and Process Descriptions. In Proceedings of International Workshop on Rule Markup Languages for Business Rules on the Semantic Web, Jun Volker Haarslev, Ralf Möller, and Michael Wessel. The Description Logic AL- CNHR+ Extended with Concrete Domains: A Practically Motivated Approach. In Proceedings of International Joint Conference on Automated Reasoning, IJ- CAR 2001, R. Goré, A. Leitsch, T. Nipkow (Eds.),Siena, Italy, Springer-Verlag, Berlin., pages 29 44, Jun I. Horrocks, U. Sattler, and S. Tobies. Practical Reasoning for Expressive Description Logics. In H. Ganzinger, D. McAllester, and A. Voronkov, editors, Proceedings of the 6th International Conference on Logic for Programming and Automated Reasoning (LPAR 99), number 1705 in Lecture Notes in Artificial Intelligence, pages Springer-Verlag, Ian Horrocks and Peter F. Patel-Schneider. The Generation of DAML+OIL. Aug The 2001 International Workshop on Description Logics. 8. Ian Horrocks and U. Sattler. Ontology Reasoning for the Semantic Web. In In B. Nebel, editor, Proc. of the 17th Int. Joint Conf. on Artificial Intelligence (IJ- CAI 01), Morgan Kaufmann, pages , C. Lutz. NExpTime-complete Description Logics with Concrete Domains. LuFG Theoretical Computer Science, RWTH Aachen, Germany, Jeff Z. Pan. Web Ontology Reasoning in the SHOQ(Dn) Description Logic. In Proceedings of the Methods for Modalities 2 (M4M-2), Nov ILLC, University of Amsterdam, URL panz/zhilin/download/paper/ Pan-shoqdn-2001.pdf. 11. Jeff Z. Pan and Ian Horrocks. Metamodeling Architecture of Web Ontology Languages. In Proceeding of the Semantic Web Working Symposium (SWWS), July URL panz/zhilin/download/paper/ Pan-Horrocks-rdfsfa-2001.pdf. 12. Frank van Harmelen, Peter F. Patel-Schneider, and Ian Horrocks. A Model- Theortis Semantics of DAML+OIL (March 2001). Mar